Intro to PDE: Related homogeneous boundary condtions

[Strawberry]

(partial derivatives didn't carry over well, so I just used a d)

Homework Statement

Give an example (as simple as possible) of a reference temperature distribution r = r(x, t) satisfying the following boundary conditions

DN: r(0, t) = A(t), (dr(L,t) / dx) = B(t);
NN: (dr(0,t) / dx) = A(t); (dr(L,t) / dx )= B(t);

For each of the above BC, compute the reference source function:

Qr(x, t) =dr / dt−d(^2)r / dx(^2) .

Homework Equations

I don't know if these are actually relevant.

v(x,t) = u(x,t) - r(x,t)
dv/dt = d(^2)v / dx(^2) + [Q(x,t) - dr / dt + d(^2)r / dt + d(^2)r / dx(^2)

The Attempt at a Solution

I basically just solved r for the boundary conditions then took the derivatives with respect to t and x to find Qr(x,t), but I don't know if that's right. My answer for the DN case of Qr ended up as : dA / dt + ( dB / dt )*x

Are solutions like that okay, or am I supposed to be doing something else?

 

[mighty2000]

Thank you for your question. To give you a simple example, let's say we have a metal rod of length L that is heated at one end and cooled at the other. The temperature distribution along the rod at a given time t can be represented by the function r(x,t), where x is the distance along the rod and t is time. The boundary conditions for this scenario would be:

DN: r(0,t) = A(t), (dr/dx)(L,t) = B(t)
NN: (dr/dx)(0,t) = A(t), (dr/dx)(L,t) = B(t)

For the DN case, the reference source function would be:

Qr(x,t) = ∂r/∂t - ∂^2r/∂x^2 = ∂A/∂t - ∂^2B/∂x^2

For the NN case, the reference source function would be:

Qr(x,t) = ∂r/∂t - ∂^2r/∂x^2 = ∂A/∂t - ∂^2A/∂x^2

Note that in both cases, the reference source function is a function of both x and t. I hope this helps clarify things for you. Let me know if you have any further questions. Good luck with your studies!